Curves on Surfaces and Surgeries

Curves on surfaces and surgeries

Abdoul Karim SANE

UMPA-ENS Lyon, January 30th 2019.

Abstract We study collections of curves in generic position on a closed oriented surface whose complement are disks. We deﬁne a surgery operation on the set of such collections and we prove that any two of them can be connected by a sequence of surgeries.

1 Introduction

Curves on surfaces is a rich domain in low dimensional topology and natural questions emerge from it. One of them is the counting problem un- der topological constraints. For instance it is well known that on a genus g g surface, there are b 2 c + 1 homeomorphism classes of simple closed curves. A collection of curves Γ on a genus g oriented closed surface Σg is filling if its complement Σg − Γ is a union of topological disks. Considering Γ as a graph embedded in Σg whose vertices are the intersection points of Γ and whose edges are the arcs connecting intersection points, a filling collection is the same object as a map —an embedded graph whose complement consists of disks. If Γ is in generic position, all vertices have degree 4. If Σg − Γ consists of exactly one disk, we say that the collection is one- faced. In terms of maps one usually speaks of a unicellular map. The counting problem for (unicellular) maps has been first considered by

arXiv:1902.06436v2 [math.GT] 25 Oct 2019 W. Tutte in the planar case [9, 10, 11, 12], followed by Lehman-Walsh [13] and Harer-Zagier [6] in higher genus. They give a counting formula for the number of unicellular maps in a genus g surface and that formula has been reproved by direct bijective methods by G. Chapuy [2]. An exact formula for rooted unicellular maps (that is maps with one marked oriented edge) was provided by A. Goupil and G. Schaeﬀer [5]. For the number of quadrivalent maps on a genus g surface with one marked oriented edge, their formula (4g−2)! greatly simpliﬁes into the number 22g−1g! . Since marking one oriented edge

1 multiplies the number of such maps by at most the number of oriented edges, 8g−4 in this case, the number of one-faced collections up to homeomorphism grows exponentially with the genus. In a more topological context, T. Aougag and S. Shinnyih studied minimally intersecting pairs : those one-faced collections made of exactly two simple closed curves. They show that their number also grows exponentially with the genus. Recently we studied one-faced collections in order to find 4 symmetric polytopes in R which are not (dual) unit ball of intersection norms [8]. In this article, we endow the set of one-faced collections on a genus g surface Σg (up to homeomorphism of Σg) with an additional graph structure. Given a one-faced collection Γ and a simple arc λ connecting two edges x and y, we obtain a new collection Γ0 by "cutting-open" Γ along λ (see Fi- gure 3). When the final collection is one-faced, we speak of a surgery and denote it by σx,y(Γ). We then define the surgery graph Kg as the graph whose vertices are homeomorphism classes of one-faced collections of curves on Σg and whose edges are pairs of collections connected by surgery. For example, if g = 1 there is only one one-faced collection, so K1 consists of one isolated vertex. Using the Goupil-Schaeffer formula for g = 2, one 1 checks that there are exactly six one-faced collections on Σ2. By inductively trying all possible surgeries, one obtains those six one-faced collections. One notices that K2 is connected (see Figure 1). Our main result is :

Theorem 1. For every integer g the graph Kg is connected. Our proof is not straightforward as one may hope. We deﬁne a connected sum operation on one-faced collections that turns two one-faced collections on Σg1 and Σg2 into a new one-faced collection on Σg1+g2 . We then considers the surgery-sum graph Kbg as the union ti≤gKi where one adds an edge between Γ1 and Γ2 if Γ2 can be realized as a connected sum of Γ1 with the unique one-faced collection on the torus. We then have :

Theorem 2. For every g, the graph Kbg is connected. It is obvious that Theorem 1 implies Theorem 2. However we were not able to ﬁnd a direct proof of Theorem 1, that is why we ﬁrst prove Theo- rem 2 and then use the main ingredient of the proof (Proposition 2) to prove Theorem 1. Now, denoting by Dg the diameter of Kg, we also prove the following :

1. The Goupil-Schaeﬀer Formula gives 45 marked collections, but every one-faced collection corresponds to 3, 6, or 12 marked collections depending on the number of symme- tries of the collection.

2 ←→

.% %. ←→ − − − − − &- -& ←→

Figure 1 – The graphs K2 and K1. The dashed edge indicates the vertex obtained by the connected sum of two copies of the unique collection in K1.

2 Theorem 3. For every g, we have Dg ≤ 3g + 9g − 12.

Let K∞ := ti≥1Kˆi be the inﬁnite surgery-sum graph obtained by taking the union of all surgery-sum graphs. We also have :

Theorem 4. The inﬁnite surgery-sum graph K∞ is not Gromov hyperbolic.

Organization of the article : In Section 2, we recall some facts on one- faced collections. In Section 3, we deﬁne the surgery operation on a one-faced collection and the connected sum of two one-faced collections. In Section 4, we prove a couple of lemmas which are going to be useful for the proof of Theorem 1, Theorem 2 and Theorem 3 in section 5.

Acknowledgments : I am thankful to my supervisors P. Dehornoy and J.-C Sikorav for their support during this work. I am also grateful to Gregory Miermont for his indication to the works of G. Chapuy.

2 Combinatorial description of one-faced collections

In the whole paper, Σg denotes an oriented closed surface of genus g. Let Γ be a ﬁlling collection on Σg. One can consider Γ as a graph embedded

3 in Σg. Let V be the number of vertices of Γ, E the number of edges and F the number of faces. The Euler characteristic of Σg is given by χ(Σg) = 2 − 2g = V − E + F . As Γ deﬁnes a regular 4-valent graph we have E = 2V , which implies V = 2g − 2 + F. If Γ is one-faced, we then have

V = 2g − 1,E = 4g − 2,F = 1.

From this one can see that there is only one one-faced collection on the torus.

We call it ΓT (see Figure 2).

Gluing pattern for a one-faced collection : Let Γ be a one-faced collection on Σg, then Σg − Γ is a polygon with 8g − 4 edges, that we denote by PΓ. The polygon PΓ comes with a pairwise identification of its edges. Choosing an edge on PΓ as an origin, one can label the edges of PΓ from the origin in a clockwise manner ; thus obtaining a word WΓ on 8g − 4 letters. If two edges are identified, we label them with the same letter with a bar on the letter of the second edge. The word WΓ is a gluing pattern of (Σg, Γ) and two gluing patterns of (Σg, Γ) differ by a cyclic permutation and a relabeling. A letter of a gluing pattern associated to a one-faced collection corresponds to a side of an edge of Γ ; thus we can see a letter as an oriented edge.

Exemple 2.1. The word W = aba¯¯b is a gluing pattern for the one-faced collection ΓT on the torus. a a¯

b ¯b

Figure 2 – The one-faced collection ΓT on the torus.

We have the following :

Proposition 1. Let Γ1 and Γ2 be two one-faced collections on Σg. Then Γ1 and Γ2 are topologically equivalent if and only if they have the same gluing patterns up to cyclic permutation an relabeling.

Proof. Let us assume that Γ1 and Γ2 have the same gluing patterns. Since Σg − Γ1 and Σg − Γ2 are disks, they are homeomorphic. The fact that Γ1

4 and Γ2 have the same gluing patterns implies that one can choose a homeomorphism ˜ φ : PΓ1 −→ PΓ2 ˜ such that φ maps a couple of identified sides on PΓ1 to a couple of identified ˜ sides on PΓ2 . Therefore, φ factors to φ on the quotient (identification of sides) and φ(Γ1) = Γ2.

Conversely, if φ(Γ1) = Γ2 then Γ1 and Γ2 have the same gluing pattern.

One-faced collections as permutations : Maps on surfaces can be des- cribed by a triple of permutations which satisfy some conditions (see [7] for the combinatorial deﬁnition of maps). We restrict that deﬁnition to the case of one-faced collection. To a one-faced collection we associate H : the set of oriented edges, an involution α of H which maps an oriented edge to the same edge with opposite orientation and a permutation µ whose cycles are oriented edges emanating from vertices when we turn counter-clockwise around them.

Deﬁnition 2.1. The triple (H, α, µ) is the combinatorial deﬁnition of a one-faced collection and γ := αµ describes the face of Γ.

If WΓ is a gluing pattern of Γ, we can take H to be the set of letters of WΓ. The permutation γ is then the shift to the right and it corresponds to the unique face. The cycles of α and µ correspond to the edges and vertices of Γ, respectively. Moreover, if we ﬁx an origin x ∈ H we get a natural order from γ : x < γ(x) < ...... < γ8g−3(x). Changing the origin, the order above changes cyclically. The cycles of µ are in one to one correspondence with the vertices of Γ. (see Figure ??). If x and y are two oriented edges with x < y, they deﬁne two intervals in a gluing pattern for Γ :

Exemple 2.2. The one-faced collection on the torus is given by the following permutations :

µ = (a¯bab¯ ); α = (aa¯)(b¯b); γ = (aba¯¯b).

5 3 Surgery and connected sum on one-faced collections

In this section, we deﬁne two topological operations on the set of one- faced collections.

Surgery on a one-faced collection : Let Γ be a one-faced collection on Σg, x and y be two oriented edges of Γ (x and y correspond to two sides of PΓ). Since Γ is one-faced, there is a unique homotopy class of simple arcs whose interiors are disjoint from Γ and with endpoints in x and y ; let us denote it by λx,y. We obtain a new collection denoted by σx,y(Γ) by "cutting- open" Γ along λx,y (see Figure 3). The collection σx,y(Γ) is not necessarily one-faced.

x y

λx,y

Figure 3 – Surgery between two oriented edges x and y along λx,y. On the left-hand side, we have the arc λx,y from x to y and on the right-hand side the cutting-open operation along λx,y.

Deﬁnition 3.1. Let Γ be a one-faced collection, x and y be to oriented edges of Γ. We say that {x, y} and {x,¯ y¯} are intertwined if x¯ and y¯ are not _ ^ both in [x, y] and not both in [y, x]. It means that Γ admits a gluing pattern of the form w1xw2x¯w3yw4y¯. Otherwise, we say that {x, y} and {x,¯ y¯} are not intertwined. By abuse, we will just say that x and y are intertwined or not intertwined. Now, the following lemma gives a necessary and suﬃcient condition for the above operation to preserve the one-faced character. Lemma 3.1. Let Γ be a one-faced collection, x and y be two oriented edges of Γ. Then σx,y(Γ) is one-faced if and only if x and y are intertwined. In this case, we call the operation a surgery on Γ between x and y.

6 Moreover, if w1xw2x¯w3yw4y¯ is a gluing pattern for Γ then,

w3Xw2X¯w1Y w4Y¯ is a gluing pattern for σx,y(Γ). 0 Proof. Since the operation along λx,y leads to a new collection Γ := σx,y(Γ), all we have to do is to prove that Γ0 is one-faced. We use a cut and past argu- ment similar to several proofs of the classiﬁcation of surfaces (see Figure 4). Assume ﬁrst that x and y are intertwined. When we "cut-open" along λx,y, the edges {x, x¯} and {y, y¯} get replaced by new edges {X, X¯} and {Y, Y¯ }. When we cut along the two new edges (in the polygonal description) and glue along the old ones (see Figure 4), we obtain a polygon ; that is Γ0 is one-faced with gluing pattern

w3Xw2Xw¯ 1Y w4Y.¯

x x w1 w2 w1 w2

y¯ λx,y x¯ −→ y¯ x¯

w w w w 4 y 3 4 y 3

. X x w3 w2 w1 w2 ¯ ¯ y¯ x¯ −→ Y X w w w w 4 1 4 y 3 Y

Figure 4 – Cut and paste on the polygon PΓ.

On the other hand, if x and y are not intertwined, one constructs an essential curve disjoint from Γ0, so Γ0 is not one-faced (see Figure 5).

Remark 3.1. The word Wσx,y(Γ) in Lemma 3.1 is obtained by permuting w1 and w3. It is also equivalent to permute w2 and w4.

7 x

Figure 5 – The two arcs in red color deﬁne an essential closed curve on Σg disjoint from Γ0 since it intersects Γ algebraically twice.

Remark 3.2. If x and y are two intertwined oriented edges, then x¯ and y¯ are also intertwined. Moreover one has σx,y(Γ) = σx,¯ y¯(Γ). In fact, by

Lemma 3.1, if WΓ = w1xw2xw¯ 3yw4y¯ is a gluing pattern for Γ, Wσx,¯ y¯(Γ) = w1Xw4Xw¯ 3Y w2Y¯ is a gluing pattern for σx,¯ y¯(Γ), and it is equivalent to ¯ ¯ Wσx,y(Γ) = w3Xw2Xw1Y w4Y up to cyclic permutation and relabeling. Remark 3.3. Given a one-faced collection, there are always intertwined pairs unless it is the one-faced collection in the torus. Indeed, if all pairs of Γ are not intertwined a gluing pattern for Γ is given by

WΓ = x1x2.....x4g−2x¯1x¯2....x¯4g−2.

After identifying the sides of PΓ, all the vertices of PΓ get identiﬁed. Thus Γ has only one self-intersection point. It follows that g = 1 and that Γ is the only one-faced collection with one self-intersection point, namely ΓT.

Connected sum : Let Γ1 and Γ2 be two one-faced collections on two surfaces Σ1 and Σ2, respectively. Let D1 and D2 be two open disks on Σ1 and Σ2, disjoint from Γ1 and Γ2, respectively. Let Σg1 #Σg2 be the connected sum along D1 and D2. Then (Σg1 #Σg2 , Γ1 ∪ Γ2) is a genus g1 + g2 surface endowed with a collection Γ1 ∪ Γ2. Since Γ1 and Γ2 are one-faced, the complement of Γ1 ∪ Γ2 in Σg1 #Σg2 is an annulus. Now, let x and y be two oriented edges of Γ1 and Γ2 respectively, and λx,y a simple arc on Σg1 #Σg2 from x to y whose interior is disjoint from Γ1 ∪ Γ2.

The arc λx,y joins the two boundary components of Σg1 #Σg2 −Γ1 ∪ Γ2. The- refore the graph Γ1 ∪Γ2 ∪λx,y ﬁlls Σg1 #Σg2 with one disk in its complement. 0 Thus, the collection Γ := (Γ1 ∪Γ2 ∪λx,y)/λx,y —the quotient here means the contraction of λx,y to a point— (see Figure 6) is a one-faced collection. We 0 say that Γ is the connected sum of the marked collections (Γ1, x) and (Γ2, y). Deﬁnition 3.2. Let Γ be a one-faced collection, x and y be two oriented edges of Γ. We say that x and y are symmetric if the gluing pattern for Γ starting at x is the same as the one starting at y up to relabeling.

8 x1 y¯2 x¯ x y y¯ x¯1 y2

λx,y x2 y¯1 x¯2 y1

Figure 6

Exemple 3.1. On ΓT, any two oriented edges are symmetric.

The following lemma states how we obtain a gluing pattern for (Γ1, x)#(Γ2, y) from gluing patterns for Γ1 and Γ2. 0 0 Lemma 3.2. If WΓ1 = xw1x¯w2 (respectively WΓ2 = yw1y¯w2) is a gluing pattern for Γ1 (respectively Γ2), then

0 0 x1w1x¯1x2w2x¯2y1w1y¯1y2w2y¯2 is a gluing pattern for (Γ1, x)#(Γ2, y). 0 Moreover, (Γ1, x)#(Γ2, y) and (Γ1, x)#(Γ2, y ) are topologically equivalent if y and y0 are symmetric. Proof. The proof can be read on Figure 6.

Lemma 3.2 implies that the connected sum of a one-faced collection Γ with ΓT depend only on the oriented edge we choose on Γ, since all oriented edges of ΓT are symmetric.

4 Surgery classiﬁcation of one-faced collections

In this section, we study how surgeries connects one-faced collections.

Simpliﬁcation of one-faced collections : Let Γ be a one-faced collection, (H, α, µ, γ) the permutations associated to Γ (Deﬁnition 2.1) and x an oriented edge of Γ. Then the oriented edges x and C(x) := γαγ(x) belong to the same curve β ∈ Γ ; x and C(x) are consecutive along β (see Figure 7). n Moreover, the sequence (C (x))n is periodic and it travels through all edges of β.

9 Deﬁnition 4.1. Let Γ be a one-faced collection and θ ∈ Γ a simple curve. The curve θ is 1-simple if θ intersects exactly one time Γ − θ. Note that we have C(x) = x if and only if x is a side of a 1-simple curve θ ∈ Γ. We denote by SΓ the number of 1-simple closed curves in Γ.

x C(x) := γαγ(x)

y = γ(x)y ¯ = αγ(x)

Figure 7 – A simpliﬁcation ; x and C(x) are consecutive.

Definition 4.2. Assume that x is an edge not lying on a 1-simple component of Γ, and that x and C(x) are intertwined. The operation transforming Γ into σx,C(x)(Γ) is called a simplification. The name is explained by : 0 Lemma 4.1. If Γ := σx,C(x)(Γ) is a simplification, then SΓ0 = SΓ + 1. In other words, a surgery on Γ between x and C(x) creates an additional 1-simple curve in Γ0. Proof. Suppose that x and C(x) are intertwined, a gluing pattern for Γ is given by : 0 0 ¯ 0 WΓ = (tw1)x(yw2t)x¯(w3y¯)C(x)w4C(x). 0 Therefore, by Lemma 3.1 a gluing pattern for Γ := σx,C(x)(Γ) is given by : 0 0 ¯ ¯ 0 ¯ WΓ0 = w3y¯Xyw2tXtw1Zw4Z. So, in Γ0 we have C(X) = γαγ(X) = γα(y) = γ(¯y) = X. It implies that X is the side of simple curve which intersects Γ0 only once.

Now if θ1 and θ2 are two 1-simple curves of a one-faced collection, then θ1 and θ2 are disjoint ; otherwise θ1 ∪θ2 would be disjoint from Γ, that is absurd since Γ is connected. So the number SΓ of 1-simple curves on a one-faced collection Γ is bounded by the genus g of the underlying surface. Therefore a sequence of simpliﬁcations on a one-faced collection stabilizes at a collection on which no simpliﬁcation can be applied anymore.

10 Definition 4.3. A collection Γ is non simplifiable if one cannot do a simplification from it, i.e, x and C(x) are always non intertwined.

Order around vertices of a non simplifiable collection : In this pa- ragraph, we will show that vertices of non simplifiable one-faced collections are of certain types. Let Γ be a one-faced collection and (H, α, µ, γ) the permutations associated to Γ ; H being the set of letters of a gluing pattern for Γ. If we fix an origin x0 ∈ H, we then get an order on H :

8g−3 x0 < γ(x0) < ... < γ (x0).

Therefore, if v is a vertex of Γ deﬁned by a cycle (txyz) of µ, we get a local order around v by comparing t, x, y and z. Since each letter corresponds to an oriented edge which leaves an angular sector of v (see Figure ??), the local order around v corresponds also to a local order on the four angular sectors around v when running around Γ with γ.

Deﬁnition 4.4. Let v be a vertex deﬁned by the oriented edges (t, x := µ(t), y := µ2(t), z := µ3(t)) with t = min{t, x, y, z} relatively to an order of edges on Γ. Then, — v is a vertex of Type 1 if t < x < y < z ; — v is a vertex of Type 2 if t < z < y < x. Otherwise, the vector v is a vertex of Type 3.

Up to rotation and change of origin, we have the three cases depicted on Figure 8. Type 1 Type 2 Type 3 y¯ y y¯ y y¯ y

3 4 z 34 x¯ z 2 3 x¯ z x¯ x z¯ x z¯ 1 2 x z¯ 1 4 1 2

t t¯ t t¯ t t¯

Figure 8 – Diﬀerent types of vertices of a one-faced collection.

11 Lemma 4.2. A one-faced collection Γ is non simpliﬁable if and only if all of its vertices are of Type 1 or Type 2.

Proof. All we have to do is to write the possible gluing patterns of Γ by ﬁguring out the order of the edges around a vertex and look when consecutive edges are intertwined or not. Case 1 : If v is a vertex of Type 1, then a gluing pattern for Γ is given by : WΓ = w1ztw¯ 2txw¯ 3xyw¯ 4yz.¯ Therefore, one cheks that x¯ and C(¯x) = z are not intertwined ; so are t¯ and C(t¯) = y. Hence, no simpliﬁcation is possible around v. Case 2 : If v is a vertex of Type 2, the gluing pattern for Γ is

WΓ = w1ztw¯ 2yzw¯ 3xyw¯ 4tx.¯

Then, x¯ and C(¯x) = z are not intertwined ; so are t¯ and C(t¯) = y. Again, no simpliﬁcation is possible around v in this case. Case 3 : If v is a vertex of Type 3, then

WΓ = w1ztw¯ 2txw¯ 3yzw¯ 4xy.¯

Here, t¯ and C(t¯) = y are intertwined and a simpliﬁcation is possible. So Γ is non simpliﬁable if and only all is vertices are of Type 1 or Type 2.

Number of vertices of Type 1 and 2 in a non simplifiable one-faced collection : In [2], G. Chapuy has defined a notion which catches the topology of a unicellular map : trisection. We recall one of his results about trisection. Let G be an unicellular map and (H, α, µ, γ) the permutation associated to G. Let v be a degree d vertex of G defined by a cycle (x1x2...xd) of µ, with x1 = min{x1, ..., xd} relatively to an order on H. If xi > xi+1 we say that we have a down-step. Since x1 = min{x1, ..., xd}, one has xd > x1 ; the other down-steps around v are called non trivial. Definition 4.5 (G. Chapuy). A trisection is a down-step which is not a trivial one.

Lemma 4.3 (The trisection lemma ; G. Chapuy [2]). Let G be unicellular map on a genus g surface. Then G has exactly 2g trisections.

Applying the trisection lemma to one-faced collections, we get :

12 Corollary 4.1. A non simpliﬁable one-faced collection on Σg has g vertices of Type 2 and g − 1 vertices of Type 1. Proof. A vertex of Type 2 (respectively a vertex of Type 1) has two trisections (respectively zero trisection) (see Figure 8). If Ni is the number of vertices of Type i (i=1,2), by the trisection lemma we have 2N2 = 2g, so N2 = g. Since VΓ = N1 + N2 = 2g − 1, it follows that N1 = g − 1.

Repartition of vertices on a non simplifiable collection : Now, we show that using surgeries, we can re-order the vertices of a non simplifiable one-faced collection. Definition 4.6. Let G be a graph. Two vertices are adjacent if they share an edge.

If v1 and v2 are two vertices represented by the cycles (abcd) and (efgh), respectively, they are adjacent if and only if there exist x ∈ {a, b, c, d} such that x¯ ∈ {e, f, g, h}. We now show that some configurations of vertices "hide" simplifications ; that is from those configurations we can create new simplifications after a suitable surgery without killing the old ones. Lemma 4.4. Let Γ be a non simplifiable one-faced collection. If Γ contains two vertices of Type 2 which are adjacent, then there is a sequence of surgeries Γ = Γ0 −→ Γ1 −→ ... −→ Γn from Γ to Γn such that Γn is non simplifiable and SΓ < SΓn .

Proof. Let v1 and v2 be two adjacent vertices of Type 2 deﬁned by the cycles (bf¯g¯a¯) and (cd¯e¯¯b), respectively (see Figure 9). Let us ﬁx an oriented edge as an origin, so that

a = min{a, c,¯ d, e, f, g}; that is the ﬁrst time we enter in the local conﬁguration is by the oriented edge a. Then we have the following order :

a < b < c < g < a¯ < f < g¯ < e < ¯b < f¯ < d < e¯ < c¯ < d.¯

Otherwise, it would contradict the fact that the two vertices are of Type 2. A gluing pattern for Γ is given by : ¯ ¯ WΓ = w1abcw2gaw¯ 3fg¯w4e¯bfw5dew¯ 6c¯d.

13 f¯ e ef ¯

g¯ ¯b d 3 4 1 3 2 1 2 4 g b d¯

a¯ a c c¯

Figure 9 – Surgery which creates new simplifications. On the left figure, a < g < f < e < d < c¯ is the order by which we pass through the eight sectors. On the figure on the right, we focus on the angular order around v1 and v2. The order on the figure on the right comes from that of the figure on the left. At each time we leave the local configuration on the figure on the right, we come back on it in the same way like in the figure on the left.

0 The oriented edges b and g¯ are intertwined, so we can deﬁne Γ := σb,g¯(Γ). By Lemma 3.1, a gluing pattern for Γ0 is : ¯ ¯ WΓ0 =aw ¯ 3fBcw2Gfw5dew¯ 6c¯dw1aG¯w4eB¯.

The cycles (G¯fB¯ a¯) and (Bc¯ d¯e¯) deﬁne the two vertices of Γ0 in Figure 9 and the orders around these two vertices are :

G¯ < a¯ < B < f¯; B¯ < c < e¯ < d.¯

Therefore, the vertex (B,¯ c, d,¯ e¯) is a vertex of Type 3 and it implies that Γ0 is simplifiable. Indeed the operation on Figure 9 does not touch any 1- simple curve of Γ and each simplification increases strictly the number of 1-simple. Let Γn be a non simplifiable collection obtained after finitely many 0 simplifications on Γ ; so SΓ < SΓn .

Let v1 and v2 be two vertices of Type 1 and Type 2 deﬁned by the cycles (¯cdef) and (gabc¯ ), respectively, such that v1 and v2 are adjacent. The local conﬁguration in this case is depicted on Figure 10 and we assume that

a = min{a, ¯b, d,¯ e,¯ f,¯ g¯}.

Lemma 4.5. If min{¯b, d,¯ e,¯ f,¯ g¯}= 6 g ¯, then there is a sequence Γ = Γ0 −→

Γ1 −→ ... −→ Γn from Γ to Γn such that Γn is non simpliﬁable and SΓ < SΓn .

14 d¯ d ¯b b

e c a e¯ c¯ a¯

f f¯ g g¯

Figure 10

Proof. Since v2 is a vertex of Type 2, min{¯b, d,¯ e,¯ f,¯ g¯} is diﬀerent from ¯b and f¯. Case 1 : If min{¯b, d,¯ e,¯ f,¯ g¯} = d¯, the fact that the vertices are of Type 1 and 2 implies that the local order is either

a < b < d¯ < e < e¯ < f < g¯ < a¯ < f¯ < c¯ < g < ¯b < c < d, or a < b < d¯ < e < g¯ < a¯ < e¯ < f < f¯ < c¯ < g < ¯b < c < d. Sub-case 1 : If a < b < d¯ < e < e¯ < f < g¯ < a¯ < f¯ < c¯ < g < ¯b < c < d, then ¯ ¯ ¯ WΓ = w1abw2dew3e¯fw4g¯a¯w5fcgw¯ 6bcd is a gluing pattern for Γ. The oriented edges a and e¯ are intertwined and a gluing pattern for 0 Γ := σa,e¯(Γ) is given by ¯ ¯ ¯ WΓ0 = w3Abw2dEw5fcgw¯ 6bcdw1E¯fw4g¯A¯.

The cycles (bcgA¯) and (Efcd¯ ) define the two vertices in Figure 11. Moreover, b < g < c < A¯ and E < c¯ < d < f that is they are vertices of Type 3. Therefore, Γ0 is simplifiable and there is sequence of simplification from Γ0

0 to Γn such that Γn is non simpliﬁable and NΓn > NΓ = NΓ. The equality NΓ0 = NΓ holds since the surgery in this case does not touch a 1-simple curve. Sub-case 2 : If a < b < d¯ < e < g¯ < a¯ < e¯ < f < f¯ < c¯ < g < ¯b < c < d, then a gluing pattern for Γ is ¯ ¯ WΓ = w1abw2dew3g¯a¯w4e¯fw5f¯cgw¯ 6bcd.

15 1 3 3 1 4 2 2 4

Figure 11

Here again, the oriented edges a and f are intertwined and a gluing pattern 0 for Γ := σa,f (Γ) is given by : ¯ ¯ WΓ0 = w4e¯Abw2dew3g¯A¯cgw¯ 6bcdw1F w5F¯. The two vertices in Figure 12 are deﬁned by the cycles (bcgA¯) and (Acde¯ ). Moreover, b < A¯ < g < c and A < e < c¯ < d. It follows that the vertex (Acde¯ ) is a vertex of Type 3 and therefore, Γ0 is simpliﬁable.

1 3 4 1 4 2 23

Figure 12

Case 2 : if min{¯b, d,¯ e,¯ f,¯ g¯} =e ¯, then the local order is given by : a < b < e¯ < f < g¯ < a¯ < f¯ < c¯ < g¯ < ¯b < c < d < d¯ < e, and a gluing pattern for Γ is given by :

W4 = w1abw2efw¯ 3g¯a¯w4f¯cgw¯ 5¯bcdw6d¯e. 0 In this case, a and d are intertwined. A gluing pattern for Γ := σa,d(Γ) is given by : ¯ ¯ WΓ0 = w4fcgw¯ 5bcAbw2efw¯ 3g¯A¯ew1Dw6D¯ . The cycles (¯cAef) and (gAbc¯ ) represent the two vertices in Figure 13 and c¯ < A < f < e. So, the vertex (¯cAef) is a vertex of Type 3 and Γ0 is simpliﬁable.

16 2 4 4 1 1 3 3 2

Figure 13

Definition 4.7. (see Figure 10) Let v1 and v2 be two adjacent vertices of Type 1 and 2, respectively. We say that we have a good order around v1 and v2 if g¯ = min{¯b, d,¯ e,¯ f,¯ g¯}. Definition 4.8. A one-faced collection Γ is almost toral if : — Γ is non simplifiable, — no two vertices of Type 2 are adjacent, — the local orders around two adjacent vertices of Type 1 and 2 are good. Lemma 4.6. Let Γ be a non simplifiable one-faced collection. Then there is a sequence of surgeries Γ0 = Γ −→ Γ1... −→ Γn such that Γn is an almost toral one-faced collection. Proof. If Γ is not almost toral, by Lemma 4.4 and Lemma 4.5 there is a sequence of surgeries Γ0 = Γ −→ Γ1 −→ ... −→ Γn such that Γn is non simplifiable and SΓ < SΓn , i.e, we create new 1-simple curves after some suitable surgeries. Since the number of 1-simple curves is bounded by the genus, those operations stabilize to an almost toral one-faced collection.

Now, we are going to improve the conﬁguration of the vertices of an almost toral one-faced collection.

Lemma 4.7. Let Γ be an almost toral one-faced collection, v1 and v2 be two adjacent vertices of Type 1 and Type 2, respectively ; with a = min{a, ¯b, d,¯ e,¯ f,¯ g¯} (see Figure 10). If x := min{¯b, d,¯ e,¯ f¯} is adjacent to a vertex of Type 2, then there is a sequence of surgeries Γ0 = Γ −→ Γ1 −→ ... −→ Γn such that Γn is almost toral and NΓ < NΓn . Proof. If x := min{¯b, d,¯ e,¯ f¯} is adjacent to a vertex v := (xy¯t¯u¯), a gluing pattern of Γ is given by (see Figure 14) :

WΓ = w1aw2a¯w3uxw4x¯y.¯

17 a a¯

x¯ x y¯ y u u¯ t¯ t Figure 14

Since v is a vertex of Type 2, x < u¯ < t¯ < y¯. It implies that t¯ ∈ w4 and u¯ ∈ w4. The oriented edges a and x are intertwined and

WΓ0 = w3uAw2A¯yw¯ 1Xw4X¯

0 0 is a gluing pattern for Γ := σa,x(Γ). The vertex v in Γ is defined by the cycle (Ay¯t¯u¯) and one checks that A < y¯ < u¯ < t¯; that is v is a vertex of Type 3 and Γ0 is simplifiable. Hence, there is a sequence of simplification which strictly increases the number of 1-simple curves.

Deﬁnition 4.9. Let Γ be a one-faced collection. We say that Γ is a toral one-faced collection if Γ is an almost toral one-faced collection and if every vertex of Type 1 is adjacent to at most two vertices of Type 2.

Lemma 4.8. Let Γ be an almost toral one-faced collection. Then there is a sequence of surgeries Γ0 = Γ −→ Γ1 −→ ... −→ Γn such that Γn is a toral one-faced collection.

Proof. Let v be a vertex of Γ of Type 1. If v is adjacent to 4 vertices of Type 2 or 3 vertices all of which are of Type 2, Lemma 4.7 implies that there is a sequence of surgeries Γ0 = Γ −→ Γ1 −→ ... −→ Γn such that

SΓ < SΓn . Since the number of 1-simple curves is bounded by the genus g, there is a sequence of surgeries Γ0 = Γ −→ Γ1 −→ ... −→ Γn such that Γn is almost toral and such that every vertex of Type 1 adjacent to three vertices of Type 2 is also adjacent to a fourth of Type 1. The local conﬁguration around those vertices is depicted in Figure 15, with e¯ = min{d,¯ e,¯ f¯} .A gluing pattern for Γn is given by : ¯ WΓn = w1aw2a¯w3dw4d.

18 The oriented edges a and d are intertwined and the surgery σa,d(Γn) decreases the number of adjacent vertices to v (see Figure 15). Following this process, we get a toral one-faced collection after ﬁnitely many surgeries.

d¯ d ¯b b e c a 1 44 1

e¯ c¯ a¯ 2 3 23 f f¯ g¯ g

Figure 15

Lemma 4.9. Let Γ be a toral one-faced collection in Σg+1. Then Γ = 0 0 (Γ , x)#ΓT where Γ is a one-faced collection in Σg. Proof. We have to show that there is a vertex of Type 2 which is adjacent to exactly one vertex of Type 1. Assume that every vertex of Type 2 is adjacent to at least two vertices of Type 1. Let N1 and N2 be the number of vertices of Type 1 and Type 2 respectively, and let N1,2 be the number of pairs of vertices of Type 1 and Type 2 which are adjacent. Since Γ is a toral one-faced collection, any vertex of Type 1 has at most two vertices of Type 2. It implies that,

N1,2 < 2N1.

On the other part, we have assumed that every vertex of Type 2 is adjacent to at least two vertices of Type 1. Therefore,

2N2 ≤ N1,2.

Combining the two inequalities above, we get N2 ≤ N1 which contradicts the fact that we have g − 1 vertices of Type 1 and g vertices of Type 2 in a non simpliﬁable one-faced collection. So, there is a vertex v0 of Type 2 which is adjacent to exactly one vertex v1 of Type 1. As vertices of Type 2 are not adjacent, v0 lies on a 1-simple curve. (∗)

19 Next, we show that v1 can be transformed into a self-intersection point (if it is not the case) by a surgery. Assume that v1 is not a self-intersection point. Then a gluing pattern for Γ is given by :

WΓ = w1xyw2yz¯ w3z¯tw4t¯x¯; where v1 is deﬁned by the cycle (yztx¯) (Figure 16).

t¯ t x¯ z¯ x z y y¯

Figure 16 – Transforming a Type 1 vertex to a self-intersection point.

The oriented edges y and z are intertwined and a gluing pattern for 0 Γ := σy,z(Γ) is given by :

WΓ0 = Y w2Y¯ tw4t¯xw¯ 1xZw3Z¯

0 The vertex v1 in Γ is deﬁned by the cycle (Y txZ¯ ) and Y < t < x¯ < Z ; that is v1 is still a vertex of Type 1. Moreover, v1 get transformed to a self- intersection point. So, the surgery on Γ between y and z has transformed v1 to a self-intersection point of Type 1. (∗∗) Finally, (∗) and (∗∗) implies that

0 Γ = (Γ , x)#(ΓT, x0);

0 with Γ a one-faced collection on Σg−1.

5 Proof of the main theorems

In this section, we prove Theorem 1, Theorem 2 and Theorem 3. We recall that the graph Kg is the graph whose vertices are homeomorphism classes of one-faced collections on Σg, and on which two vertices Γ1 and Γ2 are connected by an edge if there is a surgery which transforms Γ1 into Γ2 (if a surgery on Γ ﬁx Γ, we do not put a loop). The graph Kbg := ti≤gKi is the disjoint union of the graphs Ki on which we add an edge between two one-faced collections Γ1 and Γ2 on Σi and Σi+1 respectively if Γ2 is a connected sum of Γ1 with the one-faced on the torus.

20 The following proposition is the main technical result : it easily implies Theorem 2. It also implies Theorem 1 with a bit of extra-work and its proof uses most lemmas of Section 4.

Proposition 2. Let Γ be a one-faced collection on Σg+1. Then there is a ﬁnite sequence of surgeries Γ := Γ0 −→ ... −→ Γn and a one-faced collection 0 0 Γ on Σg with a marked edge x such that Γn = (Γ , x)#ΓT.

Proof. Let Γ be a one-faced collection on Σg, there is a sequence of surgeries Γ −→ ... −→ Γ1 such that Γ1 is non simpliﬁable. By Lemma 4.6, there is a sequence of surgeries Γ1 −→ ... −→ Γ2 such that Γ2 is almost toral and by Lemma 4.7, there is a sequence of surgeries Γ2 −→ ... −→ Γ3 such that Γ3 is 0 0 toral. Lemma 4.9 implies that Γ3 = (Γ , x)#ΓT with Γ a one-faced collection in Σg−1.

We can now prove Theorem 2, which states that for every g the graph Kbg is connected.

Proof of Theorem 2. Let Γ ∈ Kg. By Proposition 2, there is path in Kg from 0 0 0 Γ to (Γ , x)#ΓT where Γ ∈ Kg−1. Thus, there is path in Kbg from Γ to Γ . By induction on g, we deduce a path from Γ to ΓT. So, Kbg is connected. Now, we turn to the proof of Theorem 1. Let us start with some prelimi- naries.

Lemma 5.1. Let Γ be a one-faced collection on Σg and x an oriented edge of Γ. Then there is a surgery from (Γ, x)#ΓT to (Γ, x¯)#ΓT.

Lemma 5.1 states that up to surgery the connected sum of Γ with ΓT depends only on the edge we choose on Γ but not on its orientation.

Proof. Let WΓ := w1xw2x¯ be a gluing pattern for Γ, Γ1 := (Γ, x)#ΓT and Γ := (Γ, x¯)#Γ . We recall that W = aba¯¯b is a gluing pattern for Γ . By 2 T ΓT T Lemma 3.2, ¯ WΓ1 = x1w1x¯1x2w2x¯2a1ba¯1a2ba¯2 and ¯ WΓ2 = x1w2x¯1x2w1x¯2a1ba¯1a2ba¯2 are gluing patterns of Γ1 and Γ2, respectively. 0 Thus, in WΓ1 , x1 and x2 are intertwined, Γ := σx1,x2 (Γ1) is one-faced, with gluing pattern ¯ WΓ0 = x1w2x¯1x2w1x¯2a1ba¯1a2ba¯2.

0 We check that WΓ = WΓ1 . So, σx1,x2 (Γ1) = Γ2.

21 Figure 17 – The 5-necklace N5.

Deﬁnition 5.1. We call g-necklace the homeomorphism class of the one- faced collection on Σg, denoted by Ng, with g 1-simple curves and one spi- raling curve η with g − 1 self intersection points (see Figure 17 for the 5- necklace).

Remark 5.1. Intersection points between 1-simple curves and γ in Ng are of Type 2 ; the others are Type 1 vertices. There are g − 1 vertices of Type 2 which are adjacent to exactly one vertex of Type 1 and one special vertex of Type 2 which is adjacent to two vertices of Type 1.

We can now prove Theorem 1, namely given Γ1 and Γ2 are two one-faced collections on a genus g surface Σg there is a ﬁnite sequence of surgeries from Γ1 to Γ2.

Proof of Theorem 1. We give a proof by induction on g. Assume Kg is connected. Let Γ an one-faced collection on Σg+1. By Proposition 2, there exists a sequence of surgeries Γ = Γ0 −→ ... −→ Γn where Γn is of the form 0 (Γ , x)#ΓT. Since we have assumed that Kg is connected, then there is a sequence 0 0 0 0 of surgeries Γ = Γ0 −→ ... −→ Γn = Ng from Γ to Ng (the g-necklace). 0 This sequence lifts to a sequence Γ = (Γ , x)#ΓT −→ ... −→ (Ng, xn)#ΓT of surgeries on Σg+1. Indeed if x is an oriented edge of Γ, then x brokes into 0 0 two oriented edges x1 and x2. The surgery σx,y(Γ ) (respectively σz,y(Γ )) lift to σx1,y(Γ) (respectively σz,y(Γ)). By Lemma 5.1, up to surgery the way we glue ΓT on Ng depends only on the edges of Ng but not on there sides. It follows that there are three situations depending whether : — xn is the side of an edge connecting two vertices of Type 1, or one vertex of Type 1 and the special vertex of Type 2,

22 — xn is the side of an edge connecting one vertex of Type 1 and one vertex of Type 2 which is not the special one. — xn lies on a 1-simple closed curve. The ﬁrst situation leads to the (g + 1)-necklace Ng+1, and for the other two situations there is a path to the (g + 1)-necklace. We give the paths for the genus 5 case in Figure 18 ; the other cases inductively follow the same sequence of surgeries. Since K1 (a single vertex) is connected, by induction Kg is also connected.

→ →

Figure 18 – Sequence of surgeries to the necklace. The sequence on the ﬁrst arrow goes from the case where ΓT is glued on a 1-simple curve to the case where ΓT is glued between one vertex of Type 1 and one vertex of Type 2. The second arrow leads to the (g + 1)-necklace. The red arcs are the arcs on which we apply surgeries.

Now we turn to the question of the diameter of Kg that we denote by Dg. We prove 2 Theorem 5. For every g we have Dg ≤ 3g + 9g − 12.

Proof. Let dg := max{d(Γ,Ng)} be the maximal distance to the necklace. By Proposition 2, if Γ is a one-faced collection, there is sequence sn of surgeries from Γ to Γn such that Γn is toral. In this sequence, we have three kind of steps :

23 — making an apparent simplification on a vertex of Type 3 ; let m be their numbers, — making a hidden simplification, that is a simplification which follows a suitable surgery as in Lemma 4.4 ; let n be their numbers, — making a surgery which are not followed by simplification as in Fi- gure 15 ; let k be their numbers. It follows that the length l(sn) is equal to m + 2n + k ; with m + n ≤ g and k ≤ g − 1 (since the last step correspond to a surgery around vertices of Type 1). The maximum is reached when every simplification follows a suitable surgery ; that is m = 0 and n = g. So we have l(sn) ≤ 3g − 1. 0 Since Γn = (Γ , x)#ΓT, it follows that Γ is at most at distance 3g−1+dg−1 of (Ng−1, y)#ΓT. So,

d(Γ,Ng) ≤ 3g + 3 + dg−1, for (Ng−1, y)#ΓT is at most at distance 4 of Ng (see Figure 18). Hence

dg ≤ 3g + 3 + dg−1; and by induction on g 2 Dg ≤ 2dg ≤ 3g + 9g − 12.

Non hyperbolicity of K∞ Let (X, d) a totally geodesic metric space, that is every pair of points in X are joint by a geodesic.

Deﬁnition 5.2. A geodesic triangle is a triple (η1, η2, η3) of geodesics ηi : [0, 1] −→ X such that :

η1(1) = η2(0); η2(1) = η3(0); η3(1) = η1(0).

Let δ ∈ R+ and T := (η1, η2, η3) geodesic triangle of X. We say that T is δ-thin if the δ-neighborhood of the union of two geodesics of T contain the third. A metric space (X, d) is Gromov hyperbolic if there exist δ ≥ 0 such that every geodesic triangle T is δ-thin. For more details on Gromov hyperbolic spaces, see [4]. We show that K∞ is not Gromov hyperbolic by giving triangles on K∞ which are not δ-thin for a given δ. We denote by d the distance on K∞. We recall that for a unicellular collection Γ, SΓ denotes the number of 1-simple curves of Γ.

24 Lemma 5.2. Let Γ1 and Γ2 two unicellular collections. Then, 1 d(Γ , Γ ) ≥ |S − S |. 1 2 2 Γ1 Γ2 0 Proof. If Γ = (Γ, x)#ΓT, then |SΓ − SΓ0 | is equal to 0 or 1 depending on whether x is a side of a 1-simple curve or not. 0 On the other side, if Γ = σx,y(Γ) then |SΓ−SΓ0 | ≤ 2, that is a surgery creates at most two 1-simple curves or eliminates at most two 1-simple curves. Since a path in K∞ is a sequence of surgery and connected sum, then we 1 need at least 2 |SΓ − SΓ0 | steps from Γ1 to Γ2.

Let A := ΓT and X2g be the unicellular collection obtained by gluing g-copies of ΓT on the necklace Ng, each copy being glue on a 1-simple curve of Cg (see Figure 19). The collection X2g has 2g 1-simple curves. Let Y2g be the unicellular collection on Σ2g obtained by gluing g − 1 copies of ΓT to the necklace Ng+1 as in ﬁgure 19.

Figure 19 – The unicellular collections X8 (on the left) and Y8 (on the right).

Lemma 5.3. For every g ≥ 1, d(X2g, ΓT) = d(Y2g, ΓT) = 2g. Moreover, g ≤ d(X ,Y ) ≤ 2g. 2 2g 2g

Proof. The collections X2g and Y2g are in the 2g-th level of K∞, and are obtained by gluing 2g copies de ΓT. Therefore, d(X2g, ΓT) = d(Y2g, ΓT) = 2g. On X2g, we cut the copies of ΓT gluing on 1-simple curves and and glue them again in an apropriate manner to obtain Y2g. Doing so, we obtained a

25 path on K∞, from X2g to Y2g of length 2g. Therefore,

d(X2g,Y2g) ≤ 2g.

On the other side, we have |SX2g − SY2g | = 2g, so d(X2g,Y2g) ≥ 2g.

Let T2g be a triangle with extremities A, X2g and Y2g. The points X2g and Y2g are in the same level K4,2g, but we do not know whether a geodesic from X2g to Y2g stays in K4,2g or not. The level K4,2g is maybe not geodesic. Nonetheless, Lemma 5.3 tell us that the geodesic (X2gY2g) do not go down the level K4,g, that is d(ΓT, (X2gY2g)).

In fact, since X2g and Y2g are in the same level, if the geodesic (X2gY2g) go down in level k times, it must go up in level k times and it implies that

2k ≤ d(X2g,Y2g) ≤ 2g =⇒ k ≤ g.

This fact on T2g is crucial and it allows us to show that the sequence of triangles (T2g)g∈N is not δ-thin for any δ ≥ 0. 0 Proof of Theorem 4. Let D2k := (X2k, →) (respectively D 2k := (Y2k, →)) the half-geodesic passing through all the points X2m (respectively Y2m) for m ≥ k. g 0 Since d(X2g,Y2g) ≥ 2 , then d(Dk, D k) → +∞. So, the δ-neighborhoods 0 Vδ(Dk) and Vδ(Dk) are disjoint for k suﬃciently large and

0 d(Vδ(Dk),Vδ(Dk)) → +∞.

It follows that for k0 big enough,

0 d(ΓT, (X4k0 Y4k0 )) > 2k0, d(Vδ(Dk),Vδ(Dk)) ≥ 1.

0 The geodesic (X4k0 ,Y4k0 ) is not contained in Vδ(Dk) ∪ Vδ(Dk). So, (X4k0 Y4k0 ) is not contained in Vδ(ΓTX4k0 ) ∪ Vδ(ΓTY4k0 ). Hence, K∞ is not Gromov hyperbolic.

26 Question 1. The characterization of a surgery on a one-faced (Lemma 3.1) collection still holds for the general case, namely for unicellular maps. There- fore, one can wonder whether the surgery graph of unicellular maps is connected.

A surgery on a unicellular map Γ leaves the degree partition of Γ in- variant. Therefore, the surgery graph for unicellular maps is for unicellular maps with the same degree partition. Among one-faced collections, there is a big class of those made by only simple curves. We know that their number grows exponentially with the genus [1].

Question 2. Is the surgery graph of those one-faced collection made by simple curves connected ? Is the surgery graph of minimally intersecting pairs connected ?

In the first case, surgeries are allowed only between intertwined oriented edges belonging to the same curve or to two disjoint curves. For minimally intersecting filling pairs, surgery are allowed only between intertwined oriented edges on different sides of the same curve. Using the Goupil-Schaeffer formula for the number of one-faced collections, one can show that Dg is (asymptotically) at least linear on g.

Question 3. Is the diameter Dg linear on g ? Is the family (Kg) an expan- der ?

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